Strongly locally setwise homogeneous continua and their homeomorphism groups

Abstract

0. Introduction. In this paper we define strongly locally setwise homogeneous continua (s-l-s-h continua) and determine the minimal normal subgroups of the groups of all homeomorphisms of such continua. We also obtain some results about the topological structure of such spaces. Locally setwise homogeneous (1-s-h) continua were first introduced in [1], where it is shown that the groups of all homeomorphisms of 1-s-h continua are nonzero dimensional. These continua include all the compact, connected manifolds, with or without boundary, as well as the universal plane curve (i.e. Sierpinski curve), universal curve, and Hilbert cube. These examples are also s-l-s-h continua except possibly the universal curve and Hilbert cube. In [3] R. D. Anderson determines the minimal normal subgroups of the groups of all homeomorphisms of certain topological spaces with a complicated kind of setwise homogeneity and an invertibility property. These spaces include the universal plane curve, universal curve, S2, and S3. In [2] J. V. Whittaker shows that the compact manifolds, with or without boundary, are characterized by the algebraic structure of their groups of homeomorphisms, in the sense that any two compact manifolds are homeomorphic iff their groups of homeomorphisms are algebraically isomorphic. One of his main tools in this characterization is his development, in the first part of the paper, of the structure of the minimal normal subgroups of the groups of all homeomorphisms of the compact manifolds. His methods are similar to those of [3]. In [7] Ulam and von Neumann announce the theorem that the identity component of the homeomorphism group of the two-sphere is simple. Thus our results of ?3 will extend some of the results of [2], [3], and [7]. The paper is organized as follows. In ?2, we develop the notion of a near basis for a s-l-s-h continuum; in ?3, we use near bases to determine the minimal normal subgroups of the groups of homeomorphisms of s-l-s-h continua; in ?4, we study the structure of s-l-s-h continua some more; and finally in ?5, we show that various